ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralpr GIF version

Theorem ralpr 3614
Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralpr.1 𝐴 ∈ V
ralpr.2 𝐵 ∈ V
ralpr.3 (𝑥 = 𝐴 → (𝜑𝜓))
ralpr.4 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
ralpr (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralpr
StepHypRef Expression
1 ralpr.1 . 2 𝐴 ∈ V
2 ralpr.2 . 2 𝐵 ∈ V
3 ralpr.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 ralpr.4 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
53, 4ralprg 3610 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
61, 2, 5mp2an 423 1 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  wral 2435  Vcvv 2712  {cpr 3561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-sbc 2938  df-un 3106  df-sn 3566  df-pr 3567
This theorem is referenced by:  fzprval  9984
  Copyright terms: Public domain W3C validator